Riemann Surfaces of genus g with an automorphism of order p prime and p>g

The present work completes the classification of the compact Riemann surfaces of genus g with an analytic automorphism of order p (prime number) and p > g. More precisely, we construct a parameteriza- tion space for them, we compute their groups of uniformization and we compute their full automorphism groups. Also, we give affine equations for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.